// Numbas version: exam_results_page_options {"name": "Luis's copy of Differentiation: Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": ["algebraic manipulation", "calculus", "Calculus", "checked2015", "derivatives", "derivatives ", "differentiate a product", "differentiate polynomials", "differentiation", "elementary differentiation", "mas1601", "MAS1601", "polynomials", "product rule", "steps", "Steps"], "statement": "

Differentiate the following function $f(x)$ using the product rule.

", "name": "Luis's copy of Differentiation: Product rule", "metadata": {"notes": "\n \t\t

31/07/2012:

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Steps problem to be addressed via an issue. Now resolved.

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Checked calculation.OK.

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Improved prompt display.

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Clicking on Show steps does not lose any marks.

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Differentiate $f(x) = x^m(a x+b)^n$.

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$\\displaystyle \\simplify[std]{f(x) = x ^ {m} * ({a} * x+{b})^{n}}$

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$\\displaystyle \\frac{df}{dx}=\\;$[[0]]

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Clicking on Show steps gives you more information, you will not lose any marks by doing so.

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The product rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\$

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The product rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\$

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For this example:

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\$\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\$

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\$\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\$

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Hence on substituting into the product rule above we get:

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\$\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}}\$

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